Deterministic Leader Election in Anonymous Radio Networks

We consider leader election in anonymous radio networks modeled as simple undirected connected graphs. Nodes communicate in synchronous rounds. Nodes are anonymous and execute the same deterministic algorithm, so symmetry can be broken only in one way: by different wake-up times of the nodes. In which situations is it possible to break symmetry and elect a leader using time as symmetry breaker? To answer this question, we consider configurations. A configuration is the underlying graph with nodes tagged by non-negative integers with the following meaning. A node can either wake up spontaneously in the round shown on its tag, according to some global clock, or can be woken up hearing a message sent by one of its already awoken neighbours. The local clock of a node starts at its wakeup and nodes do not have access to the global clock determining their tags. A configuration is feasible if there exists a distributed algorithm that elects a leader for this configuration. Our main result is a complete algorithmic characterization of feasible configurations: we design a centralized decision algorithm, working in polynomial time, whose input is a configuration and which decides if the configuration is feasible. We also provide a dedicated deterministic distributed leader election algorithm for each feasible configuration that elects a leader for this configuration in time $O(n^2σ)$, where $n$ is the number of nodes and $σ$ is the difference between the largest and smallest tag of the configuration. We then prove that there cannot exist a universal deterministic distributed algorithm electing a leader for all feasible configurations. In fact, we show that such a universal algorithm cannot exist even for the class of 4-node feasible configurations. We also prove that a distributed version of our decision algorithm cannot exist.